1 · The essentials
The manuscript gives an exact test for when a certain kind of mathematical algebra absorbs another algebra called the Jiang–Su algebra. Here, “absorbs” means that combining the two by a tensor product leaves the original algebra unchanged up to mathematical equivalence. The test looks at an associated algebra built from sequences and asks whether it has any scalar-valued maps called characters. Under the stated assumptions, absorption occurs exactly when there are no such characters. A second result says that taking an infinite tensor power of an algebra without characters produces an algebra that absorbs the Jiang–Su algebra. The assumptions matter: the starting algebra must be nonzero, unital, separable, and complex. The main manuscript is marked as Lean-formalized.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine adding a particular attachment to a machine and finding that the resulting machine is equivalent to the original. The character criterion is like an exact diagnostic for whether that attachment changes anything, although the actual diagnostic concerns mathematical maps rather than physical parts.
Why it matters
The criterion connects Jiang–Su absorption to the absence of characters in a specific associated algebra. The source identifies this as an answer to the Kirchberg–Rørdam character question. It also identifies the infinite tensor-power result as an answer to the Dadarlat–Toms question.
2 · Going further
Let be a nonzero unital separable complex -algebra, and let denote the Jiang–Su algebra. Jiang–Su stability means that absorbs , expressed as . The main criterion states that, for every free ultrafilter on , this holds if and only if the norm central-sequence algebra of associated with has no characters. In particular, the statement is not restricted to a specially chosen free ultrafilter.
The second result concerns a different location for the no-character hypothesis: it is imposed on itself. If has no characters, its infinite minimal tensor power is -stable and contains a unital copy of . Both conclusions retain the assumptions that is nonzero, unital, separable, and complex.
The source places these statements in the context of the Kirchberg–Rørdam character question and the Dadarlat–Toms question, respectively. The abstract states the conclusions but does not describe the argument, so no proof mechanism can be extracted from the supplied text. Both results appear in the single main manuscript supplied here; no companion manuscript is described.
Useful notions
- Unital separable complex -algebras
- Characters of an algebra
- Minimal tensor products and infinite tensor powers
- Free ultrafilters on
- Norm central-sequence algebras
- Jiang–Su absorption
What was known before / what is new
Before
The source identifies two questions by name: the Kirchberg–Rørdam character question and the Dadarlat–Toms question. It gives no account of earlier partial results or methods, so the historical context here is limited to those named questions.
New
The main manuscript establishes the equivalence between Jiang–Su stability and the absence of characters in the norm central-sequence algebra, for every free ultrafilter, under the stated hypotheses. It also shows in the manuscript that the infinite minimal tensor power of every algebra in this class without characters is Jiang–Su stable and contains a unital copy of the Jiang–Su algebra. The repository summary presents these as answers to the two named questions.
Limits and assumptions
Every stated result assumes a nonzero unital separable complex -algebra. The criterion concerns characters of the norm central-sequence algebra, whereas the infinite tensor-power result assumes the starting algebra itself has no characters. The main result has a Lean formalization, and the supplied manuscript is marked Lean-formalized; the excerpt does not specify whether the formalization also covers every additional tensor-power and embedding claim. The source supplies neither argument details nor a list of remaining open questions, and makes no claim here outside the stated hypotheses.
Mini glossary
- Jiang–Su stability —
- The property of an algebra absorbing the Jiang–Su algebra , meaning .
- Character —
- A nonzero complex-valued -homomorphism from a -algebra.
- Unital —
- Having a multiplicative identity element.
- Separable —
- Having a countable subset that is dense in the norm topology.
- Infinite minimal tensor power —
- The algebra formed by taking infinitely many copies of an algebra using the minimal tensor product.
- Unital copy —
- An embedded copy of an algebra whose identity is the identity of the containing algebra.
Keywords : Jiang–Su stability · Central-sequence algebras · Infinite tensor powers
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
The Kirchberg–Rørdam character criterion
25/09/2026We prove the Kirchberg–Rørdam character criterion: for every free ultrafilter on ℕ, a nonzero unital separable complex C∗-algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters. We also prove that the infinite minimal tensor power of every nonzero unital separable complex C∗-algebra without characters is -stable and contains a unital copy of .